Expected Value in Gambling: The Formula, With Real Examples
Expected value, or EV, is the average amount a bet wins or loses per unit staked if you could make it endlessly. You calculate it by multiplying each possible outcome by its probability and adding the results. Every standard casino bet has a negative EV, which is simply the house edge written as a number: a red bet on single-zero roulette has an EV of -2.70%, so it costs 2.7 units per 100 staked on average. We checked the formula against real results, and over millions of rounds the measured EV landed within a fraction of a point of the calculated one.
Key takeaways
- Formula: EV = (probability of winning x amount won) - (probability of losing x amount lost). For bets with several outcomes, add probability x result for each one.
- Casino bets are negative: the EV of a standard casino bet equals minus its house edge. Red on single-zero roulette: -2.70%. A 2x cash-out on a 99% crash game: -1%.
- It predicts the average, not the session: expected loss = EV x total staked. 500 one-unit bets at -1% cost about 5 units on average, yet 39.4% of real 500-round crash sessions at 2x finished ahead.
- Measured vs calculated: over 4 million real crash rounds, a 2x cash-out returned an EV of -1.04% against the -1.00% the formula gives; red on 521,000 roulette spins, -2.8% against -2.70%.
- Positive EV is rare: casino games are built to be negative. The same maths decides whether any bet or offer is worth taking.
What is expected value in gambling?
Expected value is the average result of a bet if you could repeat it forever. It’s measured per unit staked, so an EV of -0.02 means you lose 2 units per 100 wagered on average. It doesn’t say what will happen on the next bet, which either wins or loses in full; it says where the results settle over many bets.
EV is the single number that ties together odds, payouts and the house edge. For any casino bet, EV = -(house edge). A game with a 97% return to player has an EV of -3% on every bet.
The expected value formula
For a bet with two outcomes:
EV = (P(win) x net win) - (P(lose) x stake)
For bets with more outcomes, the general form is to multiply every outcome by its probability and add them up: EV = sum of (probability x result). Probabilities must add up to 1, and results use the same unit, usually the stake.
Worked examples from real games
Red on single-zero roulette. 18 of 37 pockets are red. A 1-unit bet wins 1 unit with probability 18/37 and loses 1 with probability 19/37: EV = 18/37 - 19/37 = -1/37 = -2.70%.
A single number on the same wheel pays 35 units with probability 1/37 and loses 1 unit the other 36 times in 37. That’s 35/37 - 36/37, again -1/37: the same -2.70% as red, with far more variance.
A crash bet at 2x. On a game returning 99%, the chance of reaching 2x is 99% / 2 = 49.5%. EV = 0.495 x 1 - 0.505 x 1 = -0.01, or -1%.
Raise the crash target to 10x and the chance drops to 99% / 10 = 9.9% while the net win rises to 9 units, so 0.099 x 9 - 0.901 = -0.01 once more. In crash every cash-out target has the same EV; only the variance changes.
Calculated EV vs what really happened
A formula is a claim, so we checked it against the results we’ve recorded:
| Bet | Rounds | Calculated EV | Measured EV |
|---|---|---|---|
| Shuffle Crash, 2x cash-out | 4,034,226 | -1.00% | -1.04% |
| Shuffle Crash, 10x cash-out | 4,034,226 | -1.00% | -1.14% |
| Duel Crash, 2x cash-out | 1,317,332 | -0.10% | +0.01% |
| Aviator, 2x cash-out | 11,300 | -3.00% | -3.47% |
| Red on single-zero roulette | 521,165 | -2.70% | -2.80% |
Every gap is the size of the random error you’d expect for its sample and bet. For the 10x cash-out over 4 million rounds, chance alone moves the result by about 0.15 points either way; for Aviator’s 11,300 rounds, by about 0.9. Duel’s +0.01% is that same noise around a 0.1% edge, not a player advantage. Our house edge guide compares the same games’ published figures.
Turning odds into probabilities
Outside roulette and crash, the probabilities usually come from the odds you’re offered. The conversion is the implied probability: 1 / decimal odds, or 100 / (odds + 100) for positive American odds. Odds of 2.50 (or +150) imply 40%. The EV of a 1-unit bet at 2.50 is then p x 1.5 - (1 - p) x 1, and it only turns positive if the true chance is above 40%. That’s what a good EV in betting means: an estimate of the real probability that beats the one built into the odds. A bookmaker’s implied probabilities for all outcomes add up to more than 100%, and that excess is its margin, the same job the house edge does in a casino.
Why EV doesn’t predict your session
EV describes the average of many bets, while a session is a small handful of them. Over 500 one-unit crash bets at 2x, the expected loss is 5 units, but when we replayed 8,068 real 500-round sessions, 39.4% of them finished ahead. That spread around the average is variance, and it’s why a negative-EV game can feel like a winning one for a while. Our crash strategy test shows the full range of outcomes by cash-out target.
What EV does predict reliably is the cost of turnover: expected loss = EV x total staked. Double the stakes or the rounds and the expected cost doubles too, whatever the results in between.
Can gambling have a positive EV?
Not from the games themselves: the house edge makes every standard bet negative, and no staking plan changes that, as our tests of the Martingale and other betting systems on real rounds show.
A bonus can change the sum, because the casino adds money to it. Take a 100-unit bonus with a 30x wagering requirement played on a 97% RTP game: you must wager 3,000 units, which costs 3% x 3,000 = 90 units on average, so the bonus is worth about +10 before the terms. In practice maximum-bet rules, excluded games, win caps and time limits often cut that to zero or below, which is why the only honest way to judge an offer is to run the full EV sum with its real terms.
How to use EV as a player
Use EV to compare games, not to predict results. A -0.1% game costs a hundredth of a -10% game for the same stakes, so picking the least negative EV is the one lever that lowers your long-run cost. Then set your stake and session length so that the expected loss, EV times total staked, is an amount you’re comfortable spending on the entertainment.
If gambling stops being fun, GamCare and Gambling Therapy offer free, confidential help.
Frequently asked questions
What is the formula for expected value?
Expected value is the sum of every possible outcome multiplied by its probability. For a simple bet: EV = (probability of winning x net win) - (probability of losing x stake). For red on a single-zero roulette wheel with a 1-unit stake, that's (18/37 x 1) - (19/37 x 1) = -1/37, or -0.027 units per spin, a -2.70% expected value.
How to calculate the expected value of a gamble?
List every outcome, its probability and what you'd win or lose, then multiply each pair and add them up. A crash bet with a 2x cash-out on a 99% game succeeds with probability 0.495 for a net win of 1 unit and fails with probability 0.505 for a loss of 1 unit: 0.495 - 0.505 = -0.01. The result is your average profit or loss per unit staked.
How to calculate expected winnings?
Multiply your expected value per unit by the total you plan to stake. At an EV of -1%, putting 500 units through a game gives expected winnings of -5 units; at -2.7%, the same stakes give -13.5. Real results scatter around that figure, sometimes well above it, but the more you stake, the closer your total tends to get to it.
What is EV in terms of gambling?
EV, short for expected value, is how much a bet is worth on average. A positive EV means the bet would make money over many repetitions; a negative EV means it would lose. Every standard casino bet has a negative EV equal to the house edge, which is why casinos profit over time even though individual players win often.
What is a good expected value?
Any positive expected value is good in principle, because the bet makes money on average. In a casino that doesn't happen on standard games, so the realistic target is the least negative EV you can find: -0.1% at a crash game with a 0.1% edge costs a hundredth of what -10% does. A good EV in practice means choosing bets that lose slowly.
What is a good ROI for gambling?
For casino games, the long-run ROI equals the expected value, so it's negative: about -1% on a 99% crash game, -2.7% on single-zero roulette, and worse on many slots. Any positive ROI you see over a short stretch is variance, not a return you can plan on. Treat money spent gambling as the cost of entertainment, not an investment.
Set a budget before you play. Free, confidential help is available from GamCare and Gambling Therapy.